Migdal et al, Equation (64)

Percentage Accurate: 99.5% → 99.6%

Time: 8.9s

Alternatives: 9

Speedup: 2.0×

• 43.5% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{\cos th}{\sqrt{2}}\\ t_1 \cdot \left(a1 \cdot a1\right) + t_1 \cdot \left(a2 \cdot a2\right) \end{array} \end{array} \]

FPCore

(FPCore (a1 a2 th)
 :precision binary64
 (let* ((t_1 (/ (cos th) (sqrt 2.0))))
   (+ (* t_1 (* a1 a1)) (* t_1 (* a2 a2)))))

C

double code(double a1, double a2, double th) {
	double t_1 = cos(th) / sqrt(2.0);
	return (t_1 * (a1 * a1)) + (t_1 * (a2 * a2));
}

Fortran

real(8) function code(a1, a2, th)
    real(8), intent (in) :: a1
    real(8), intent (in) :: a2
    real(8), intent (in) :: th
    real(8) :: t_1
    t_1 = cos(th) / sqrt(2.0d0)
    code = (t_1 * (a1 * a1)) + (t_1 * (a2 * a2))
end function

Java

public static double code(double a1, double a2, double th) {
	double t_1 = Math.cos(th) / Math.sqrt(2.0);
	return (t_1 * (a1 * a1)) + (t_1 * (a2 * a2));
}

Python

def code(a1, a2, th):
	t_1 = math.cos(th) / math.sqrt(2.0)
	return (t_1 * (a1 * a1)) + (t_1 * (a2 * a2))

Julia

function code(a1, a2, th)
	t_1 = Float64(cos(th) / sqrt(2.0))
	return Float64(Float64(t_1 * Float64(a1 * a1)) + Float64(t_1 * Float64(a2 * a2)))
end

MATLAB

function tmp = code(a1, a2, th)
	t_1 = cos(th) / sqrt(2.0);
	tmp = (t_1 * (a1 * a1)) + (t_1 * (a2 * a2));
end

Wolfram

code[a1_, a2_, th_] := Block[{t$95$1 = N[(N[Cos[th], $MachinePrecision] / N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]}, N[(N[(t$95$1 * N[(a1 * a1), $MachinePrecision]), $MachinePrecision] + N[(t$95$1 * N[(a2 * a2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Initial Program: 99.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{\cos th}{\sqrt{2}}\\ t_1 \cdot \left(a1 \cdot a1\right) + t_1 \cdot \left(a2 \cdot a2\right) \end{array} \end{array} \]

FPCore

(FPCore (a1 a2 th)
 :precision binary64
 (let* ((t_1 (/ (cos th) (sqrt 2.0))))
   (+ (* t_1 (* a1 a1)) (* t_1 (* a2 a2)))))

C

double code(double a1, double a2, double th) {
	double t_1 = cos(th) / sqrt(2.0);
	return (t_1 * (a1 * a1)) + (t_1 * (a2 * a2));
}

Fortran

real(8) function code(a1, a2, th)
    real(8), intent (in) :: a1
    real(8), intent (in) :: a2
    real(8), intent (in) :: th
    real(8) :: t_1
    t_1 = cos(th) / sqrt(2.0d0)
    code = (t_1 * (a1 * a1)) + (t_1 * (a2 * a2))
end function

Java

public static double code(double a1, double a2, double th) {
	double t_1 = Math.cos(th) / Math.sqrt(2.0);
	return (t_1 * (a1 * a1)) + (t_1 * (a2 * a2));
}

Python

def code(a1, a2, th):
	t_1 = math.cos(th) / math.sqrt(2.0)
	return (t_1 * (a1 * a1)) + (t_1 * (a2 * a2))

Julia

function code(a1, a2, th)
	t_1 = Float64(cos(th) / sqrt(2.0))
	return Float64(Float64(t_1 * Float64(a1 * a1)) + Float64(t_1 * Float64(a2 * a2)))
end

MATLAB

function tmp = code(a1, a2, th)
	t_1 = cos(th) / sqrt(2.0);
	tmp = (t_1 * (a1 * a1)) + (t_1 * (a2 * a2));
end

Wolfram

code[a1_, a2_, th_] := Block[{t$95$1 = N[(N[Cos[th], $MachinePrecision] / N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]}, N[(N[(t$95$1 * N[(a1 * a1), $MachinePrecision]), $MachinePrecision] + N[(t$95$1 * N[(a2 * a2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Alternative 1: 99.6% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \cos th \cdot \frac{\mathsf{fma}\left(a1, a1, a2 \cdot a2\right)}{\sqrt{2}} \end{array} \]

FPCore

(FPCore (a1 a2 th)
 :precision binary64
 (* (cos th) (/ (fma a1 a1 (* a2 a2)) (sqrt 2.0))))

C

double code(double a1, double a2, double th) {
	return cos(th) * (fma(a1, a1, (a2 * a2)) / sqrt(2.0));
}

Julia

function code(a1, a2, th)
	return Float64(cos(th) * Float64(fma(a1, a1, Float64(a2 * a2)) / sqrt(2.0)))
end

Wolfram

code[a1_, a2_, th_] := N[(N[Cos[th], $MachinePrecision] * N[(N[(a1 * a1 + N[(a2 * a2), $MachinePrecision]), $MachinePrecision] / N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.6%

   \[\frac{\cos th}{\sqrt{2}} \cdot \left(a1 \cdot a1\right) + \frac{\cos th}{\sqrt{2}} \cdot \left(a2 \cdot a2\right) \]
2. Step-by-step derivation

   1. distribute-lft-out 99.6%

      \[\leadsto \color{blue}{\frac{\cos th}{\sqrt{2}} \cdot \left(a1 \cdot a1 + a2 \cdot a2\right)} \]

   2. associate-*l/ 99.7%

      \[\leadsto \color{blue}{\frac{\cos th \cdot \left(a1 \cdot a1 + a2 \cdot a2\right)}{\sqrt{2}}} \]

   3. associate-*r/ 99.7%

      \[\leadsto \color{blue}{\cos th \cdot \frac{a1 \cdot a1 + a2 \cdot a2}{\sqrt{2}}} \]

   4. fma-def 99.7%

      \[\leadsto \cos th \cdot \frac{\color{blue}{\mathsf{fma}\left(a1, a1, a2 \cdot a2\right)}}{\sqrt{2}} \]

3. Simplified 99.7%

   \[\leadsto \color{blue}{\cos th \cdot \frac{\mathsf{fma}\left(a1, a1, a2 \cdot a2\right)}{\sqrt{2}}} \]

4. Final simplification 99.7%

   \[\leadsto \cos th \cdot \frac{\mathsf{fma}\left(a1, a1, a2 \cdot a2\right)}{\sqrt{2}} \]

Alternative 2: 99.5% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \frac{\cos th}{\sqrt{2}} \cdot \left(a2 \cdot a2 + a1 \cdot a1\right) \end{array} \]

FPCore

(FPCore (a1 a2 th)
 :precision binary64
 (* (/ (cos th) (sqrt 2.0)) (+ (* a2 a2) (* a1 a1))))

C

double code(double a1, double a2, double th) {
	return (cos(th) / sqrt(2.0)) * ((a2 * a2) + (a1 * a1));
}

Fortran

real(8) function code(a1, a2, th)
    real(8), intent (in) :: a1
    real(8), intent (in) :: a2
    real(8), intent (in) :: th
    code = (cos(th) / sqrt(2.0d0)) * ((a2 * a2) + (a1 * a1))
end function

Java

public static double code(double a1, double a2, double th) {
	return (Math.cos(th) / Math.sqrt(2.0)) * ((a2 * a2) + (a1 * a1));
}

Python

def code(a1, a2, th):
	return (math.cos(th) / math.sqrt(2.0)) * ((a2 * a2) + (a1 * a1))

Julia

function code(a1, a2, th)
	return Float64(Float64(cos(th) / sqrt(2.0)) * Float64(Float64(a2 * a2) + Float64(a1 * a1)))
end

MATLAB

function tmp = code(a1, a2, th)
	tmp = (cos(th) / sqrt(2.0)) * ((a2 * a2) + (a1 * a1));
end

Wolfram

code[a1_, a2_, th_] := N[(N[(N[Cos[th], $MachinePrecision] / N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * N[(N[(a2 * a2), $MachinePrecision] + N[(a1 * a1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.6%

   \[\frac{\cos th}{\sqrt{2}} \cdot \left(a1 \cdot a1\right) + \frac{\cos th}{\sqrt{2}} \cdot \left(a2 \cdot a2\right) \]
2. Step-by-step derivation

   1. distribute-lft-out 99.6%

      \[\leadsto \color{blue}{\frac{\cos th}{\sqrt{2}} \cdot \left(a1 \cdot a1 + a2 \cdot a2\right)} \]

3. Simplified 99.6%

   \[\leadsto \color{blue}{\frac{\cos th}{\sqrt{2}} \cdot \left(a1 \cdot a1 + a2 \cdot a2\right)} \]

4. Final simplification 99.6%

   \[\leadsto \frac{\cos th}{\sqrt{2}} \cdot \left(a2 \cdot a2 + a1 \cdot a1\right) \]

Alternative 3: 56.7% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ a2 \cdot \frac{a2}{\frac{\sqrt{2}}{\cos th}} \end{array} \]

FPCore

(FPCore (a1 a2 th) :precision binary64 (* a2 (/ a2 (/ (sqrt 2.0) (cos th)))))

C

double code(double a1, double a2, double th) {
	return a2 * (a2 / (sqrt(2.0) / cos(th)));
}

Fortran

real(8) function code(a1, a2, th)
    real(8), intent (in) :: a1
    real(8), intent (in) :: a2
    real(8), intent (in) :: th
    code = a2 * (a2 / (sqrt(2.0d0) / cos(th)))
end function

Java

public static double code(double a1, double a2, double th) {
	return a2 * (a2 / (Math.sqrt(2.0) / Math.cos(th)));
}

Python

def code(a1, a2, th):
	return a2 * (a2 / (math.sqrt(2.0) / math.cos(th)))

Julia

function code(a1, a2, th)
	return Float64(a2 * Float64(a2 / Float64(sqrt(2.0) / cos(th))))
end

MATLAB

function tmp = code(a1, a2, th)
	tmp = a2 * (a2 / (sqrt(2.0) / cos(th)));
end

Wolfram

code[a1_, a2_, th_] := N[(a2 * N[(a2 / N[(N[Sqrt[2.0], $MachinePrecision] / N[Cos[th], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.6%

   \[\frac{\cos th}{\sqrt{2}} \cdot \left(a1 \cdot a1\right) + \frac{\cos th}{\sqrt{2}} \cdot \left(a2 \cdot a2\right) \]
2. Step-by-step derivation

   1. distribute-lft-out 99.6%

      \[\leadsto \color{blue}{\frac{\cos th}{\sqrt{2}} \cdot \left(a1 \cdot a1 + a2 \cdot a2\right)} \]

   2. associate-*l/ 99.7%

      \[\leadsto \color{blue}{\frac{\cos th \cdot \left(a1 \cdot a1 + a2 \cdot a2\right)}{\sqrt{2}}} \]

   3. associate-*r/ 99.7%

      \[\leadsto \color{blue}{\cos th \cdot \frac{a1 \cdot a1 + a2 \cdot a2}{\sqrt{2}}} \]

   4. fma-def 99.7%

      \[\leadsto \cos th \cdot \frac{\color{blue}{\mathsf{fma}\left(a1, a1, a2 \cdot a2\right)}}{\sqrt{2}} \]

3. Simplified 99.7%

   \[\leadsto \color{blue}{\cos th \cdot \frac{\mathsf{fma}\left(a1, a1, a2 \cdot a2\right)}{\sqrt{2}}} \]

4. Taylor expanded in a1 around 0 60.3%

   \[\leadsto \color{blue}{\frac{{a2}^{2} \cdot \cos th}{\sqrt{2}}} \]
5. Step-by-step derivation

   1. unpow2 60.3%

      \[\leadsto \frac{\color{blue}{\left(a2 \cdot a2\right)} \cdot \cos th}{\sqrt{2}} \]

   2. associate-*l* 60.3%

      \[\leadsto \frac{\color{blue}{a2 \cdot \left(a2 \cdot \cos th\right)}}{\sqrt{2}} \]

   3. associate-*r/ 60.3%

      \[\leadsto \color{blue}{a2 \cdot \frac{a2 \cdot \cos th}{\sqrt{2}}} \]

   4. associate-/l* 60.3%

      \[\leadsto a2 \cdot \color{blue}{\frac{a2}{\frac{\sqrt{2}}{\cos th}}} \]

6. Simplified 60.3%

   \[\leadsto \color{blue}{a2 \cdot \frac{a2}{\frac{\sqrt{2}}{\cos th}}} \]

7. Final simplification 60.3%

   \[\leadsto a2 \cdot \frac{a2}{\frac{\sqrt{2}}{\cos th}} \]

Alternative 4: 56.7% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \cos th \cdot \frac{a2}{\frac{\sqrt{2}}{a2}} \end{array} \]

FPCore

(FPCore (a1 a2 th) :precision binary64 (* (cos th) (/ a2 (/ (sqrt 2.0) a2))))

C

double code(double a1, double a2, double th) {
	return cos(th) * (a2 / (sqrt(2.0) / a2));
}

Fortran

real(8) function code(a1, a2, th)
    real(8), intent (in) :: a1
    real(8), intent (in) :: a2
    real(8), intent (in) :: th
    code = cos(th) * (a2 / (sqrt(2.0d0) / a2))
end function

Java

public static double code(double a1, double a2, double th) {
	return Math.cos(th) * (a2 / (Math.sqrt(2.0) / a2));
}

Python

def code(a1, a2, th):
	return math.cos(th) * (a2 / (math.sqrt(2.0) / a2))

Julia

function code(a1, a2, th)
	return Float64(cos(th) * Float64(a2 / Float64(sqrt(2.0) / a2)))
end

MATLAB

function tmp = code(a1, a2, th)
	tmp = cos(th) * (a2 / (sqrt(2.0) / a2));
end

Wolfram

code[a1_, a2_, th_] := N[(N[Cos[th], $MachinePrecision] * N[(a2 / N[(N[Sqrt[2.0], $MachinePrecision] / a2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.6%

   \[\frac{\cos th}{\sqrt{2}} \cdot \left(a1 \cdot a1\right) + \frac{\cos th}{\sqrt{2}} \cdot \left(a2 \cdot a2\right) \]
2. Step-by-step derivation

   1. distribute-lft-out 99.6%

      \[\leadsto \color{blue}{\frac{\cos th}{\sqrt{2}} \cdot \left(a1 \cdot a1 + a2 \cdot a2\right)} \]

   2. associate-*l/ 99.7%

      \[\leadsto \color{blue}{\frac{\cos th \cdot \left(a1 \cdot a1 + a2 \cdot a2\right)}{\sqrt{2}}} \]

   3. associate-*r/ 99.7%

      \[\leadsto \color{blue}{\cos th \cdot \frac{a1 \cdot a1 + a2 \cdot a2}{\sqrt{2}}} \]

   4. fma-def 99.7%

      \[\leadsto \cos th \cdot \frac{\color{blue}{\mathsf{fma}\left(a1, a1, a2 \cdot a2\right)}}{\sqrt{2}} \]

3. Simplified 99.7%

   \[\leadsto \color{blue}{\cos th \cdot \frac{\mathsf{fma}\left(a1, a1, a2 \cdot a2\right)}{\sqrt{2}}} \]

4. Taylor expanded in a1 around 0 60.3%

   \[\leadsto \cos th \cdot \color{blue}{\frac{{a2}^{2}}{\sqrt{2}}} \]
5. Step-by-step derivation

   1. unpow2 60.3%

      \[\leadsto \cos th \cdot \frac{\color{blue}{a2 \cdot a2}}{\sqrt{2}} \]

   2. associate-/l* 60.4%

      \[\leadsto \cos th \cdot \color{blue}{\frac{a2}{\frac{\sqrt{2}}{a2}}} \]

6. Simplified 60.4%

   \[\leadsto \cos th \cdot \color{blue}{\frac{a2}{\frac{\sqrt{2}}{a2}}} \]

7. Final simplification 60.4%

   \[\leadsto \cos th \cdot \frac{a2}{\frac{\sqrt{2}}{a2}} \]

Alternative 5: 56.7% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \cos th \cdot \frac{a2 \cdot a2}{\sqrt{2}} \end{array} \]

FPCore

(FPCore (a1 a2 th) :precision binary64 (* (cos th) (/ (* a2 a2) (sqrt 2.0))))

C

double code(double a1, double a2, double th) {
	return cos(th) * ((a2 * a2) / sqrt(2.0));
}

Fortran

real(8) function code(a1, a2, th)
    real(8), intent (in) :: a1
    real(8), intent (in) :: a2
    real(8), intent (in) :: th
    code = cos(th) * ((a2 * a2) / sqrt(2.0d0))
end function

Java

public static double code(double a1, double a2, double th) {
	return Math.cos(th) * ((a2 * a2) / Math.sqrt(2.0));
}

Python

def code(a1, a2, th):
	return math.cos(th) * ((a2 * a2) / math.sqrt(2.0))

Julia

function code(a1, a2, th)
	return Float64(cos(th) * Float64(Float64(a2 * a2) / sqrt(2.0)))
end

MATLAB

function tmp = code(a1, a2, th)
	tmp = cos(th) * ((a2 * a2) / sqrt(2.0));
end

Wolfram

code[a1_, a2_, th_] := N[(N[Cos[th], $MachinePrecision] * N[(N[(a2 * a2), $MachinePrecision] / N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.6%

   \[\frac{\cos th}{\sqrt{2}} \cdot \left(a1 \cdot a1\right) + \frac{\cos th}{\sqrt{2}} \cdot \left(a2 \cdot a2\right) \]
2. Step-by-step derivation

   1. distribute-lft-out 99.6%

      \[\leadsto \color{blue}{\frac{\cos th}{\sqrt{2}} \cdot \left(a1 \cdot a1 + a2 \cdot a2\right)} \]

   2. associate-*l/ 99.7%

      \[\leadsto \color{blue}{\frac{\cos th \cdot \left(a1 \cdot a1 + a2 \cdot a2\right)}{\sqrt{2}}} \]

   3. associate-*r/ 99.7%

      \[\leadsto \color{blue}{\cos th \cdot \frac{a1 \cdot a1 + a2 \cdot a2}{\sqrt{2}}} \]

   4. fma-def 99.7%

      \[\leadsto \cos th \cdot \frac{\color{blue}{\mathsf{fma}\left(a1, a1, a2 \cdot a2\right)}}{\sqrt{2}} \]

3. Simplified 99.7%

   \[\leadsto \color{blue}{\cos th \cdot \frac{\mathsf{fma}\left(a1, a1, a2 \cdot a2\right)}{\sqrt{2}}} \]

4. Taylor expanded in a1 around 0 60.3%

   \[\leadsto \cos th \cdot \color{blue}{\frac{{a2}^{2}}{\sqrt{2}}} \]
5. Step-by-step derivation

   1. unpow2 60.3%

      \[\leadsto \cos th \cdot \frac{\color{blue}{a2 \cdot a2}}{\sqrt{2}} \]

6. Simplified 60.3%

   \[\leadsto \cos th \cdot \color{blue}{\frac{a2 \cdot a2}{\sqrt{2}}} \]

7. Final simplification 60.3%

   \[\leadsto \cos th \cdot \frac{a2 \cdot a2}{\sqrt{2}} \]

Alternative 6: 39.4% accurate, 3.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;th \leq 9 \cdot 10^{+61}:\\ \;\;\;\;\frac{a2}{\frac{\sqrt{2}}{a2}}\\ \mathbf{else}:\\ \;\;\;\;a2 \cdot \frac{-1}{\sqrt{2} \cdot \frac{1}{a2}}\\ \end{array} \end{array} \]

FPCore

(FPCore (a1 a2 th)
 :precision binary64
 (if (<= th 9e+61)
   (/ a2 (/ (sqrt 2.0) a2))
   (* a2 (/ -1.0 (* (sqrt 2.0) (/ 1.0 a2))))))

C

double code(double a1, double a2, double th) {
	double tmp;
	if (th <= 9e+61) {
		tmp = a2 / (sqrt(2.0) / a2);
	} else {
		tmp = a2 * (-1.0 / (sqrt(2.0) * (1.0 / a2)));
	}
	return tmp;
}

Fortran

real(8) function code(a1, a2, th)
    real(8), intent (in) :: a1
    real(8), intent (in) :: a2
    real(8), intent (in) :: th
    real(8) :: tmp
    if (th <= 9d+61) then
        tmp = a2 / (sqrt(2.0d0) / a2)
    else
        tmp = a2 * ((-1.0d0) / (sqrt(2.0d0) * (1.0d0 / a2)))
    end if
    code = tmp
end function

Java

public static double code(double a1, double a2, double th) {
	double tmp;
	if (th <= 9e+61) {
		tmp = a2 / (Math.sqrt(2.0) / a2);
	} else {
		tmp = a2 * (-1.0 / (Math.sqrt(2.0) * (1.0 / a2)));
	}
	return tmp;
}

Python

def code(a1, a2, th):
	tmp = 0
	if th <= 9e+61:
		tmp = a2 / (math.sqrt(2.0) / a2)
	else:
		tmp = a2 * (-1.0 / (math.sqrt(2.0) * (1.0 / a2)))
	return tmp

Julia

function code(a1, a2, th)
	tmp = 0.0
	if (th <= 9e+61)
		tmp = Float64(a2 / Float64(sqrt(2.0) / a2));
	else
		tmp = Float64(a2 * Float64(-1.0 / Float64(sqrt(2.0) * Float64(1.0 / a2))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a1, a2, th)
	tmp = 0.0;
	if (th <= 9e+61)
		tmp = a2 / (sqrt(2.0) / a2);
	else
		tmp = a2 * (-1.0 / (sqrt(2.0) * (1.0 / a2)));
	end
	tmp_2 = tmp;
end

Wolfram

code[a1_, a2_, th_] := If[LessEqual[th, 9e+61], N[(a2 / N[(N[Sqrt[2.0], $MachinePrecision] / a2), $MachinePrecision]), $MachinePrecision], N[(a2 * N[(-1.0 / N[(N[Sqrt[2.0], $MachinePrecision] * N[(1.0 / a2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if th < 9e61

   1. Initial program 99.6%

      \[\frac{\cos th}{\sqrt{2}} \cdot \left(a1 \cdot a1\right) + \frac{\cos th}{\sqrt{2}} \cdot \left(a2 \cdot a2\right) \]
   2. Step-by-step derivation

      1. distribute-lft-out 99.6%

         \[\leadsto \color{blue}{\frac{\cos th}{\sqrt{2}} \cdot \left(a1 \cdot a1 + a2 \cdot a2\right)} \]

      2. associate-*l/ 99.7%

         \[\leadsto \color{blue}{\frac{\cos th \cdot \left(a1 \cdot a1 + a2 \cdot a2\right)}{\sqrt{2}}} \]

      3. associate-*r/ 99.7%

         \[\leadsto \color{blue}{\cos th \cdot \frac{a1 \cdot a1 + a2 \cdot a2}{\sqrt{2}}} \]

      4. fma-def 99.7%

         \[\leadsto \cos th \cdot \frac{\color{blue}{\mathsf{fma}\left(a1, a1, a2 \cdot a2\right)}}{\sqrt{2}} \]

   3. Simplified 99.7%

      \[\leadsto \color{blue}{\cos th \cdot \frac{\mathsf{fma}\left(a1, a1, a2 \cdot a2\right)}{\sqrt{2}}} \]

   4. Taylor expanded in a1 around 0 59.4%

      \[\leadsto \cos th \cdot \color{blue}{\frac{{a2}^{2}}{\sqrt{2}}} \]
   5. Step-by-step derivation

      1. unpow2 59.4%

         \[\leadsto \cos th \cdot \frac{\color{blue}{a2 \cdot a2}}{\sqrt{2}} \]

   6. Simplified 59.4%

      \[\leadsto \cos th \cdot \color{blue}{\frac{a2 \cdot a2}{\sqrt{2}}} \]
   7. Step-by-step derivation

      1. associate-/l* 59.5%

         \[\leadsto \cos th \cdot \color{blue}{\frac{a2}{\frac{\sqrt{2}}{a2}}} \]

      2. associate-*r/ 59.5%

         \[\leadsto \color{blue}{\frac{\cos th \cdot a2}{\frac{\sqrt{2}}{a2}}} \]

   8. Applied egg-rr 59.5%

      \[\leadsto \color{blue}{\frac{\cos th \cdot a2}{\frac{\sqrt{2}}{a2}}} \]

   9. Taylor expanded in th around 0 45.0%

      \[\leadsto \frac{\color{blue}{a2}}{\frac{\sqrt{2}}{a2}} \]

   if 9e61 < th

   1. Initial program 99.6%

      \[\frac{\cos th}{\sqrt{2}} \cdot \left(a1 \cdot a1\right) + \frac{\cos th}{\sqrt{2}} \cdot \left(a2 \cdot a2\right) \]
   2. Step-by-step derivation

      1. distribute-lft-out 99.6%

         \[\leadsto \color{blue}{\frac{\cos th}{\sqrt{2}} \cdot \left(a1 \cdot a1 + a2 \cdot a2\right)} \]

      2. associate-*l/ 99.6%

         \[\leadsto \color{blue}{\frac{\cos th \cdot \left(a1 \cdot a1 + a2 \cdot a2\right)}{\sqrt{2}}} \]

      3. associate-*r/ 99.6%

         \[\leadsto \color{blue}{\cos th \cdot \frac{a1 \cdot a1 + a2 \cdot a2}{\sqrt{2}}} \]

      4. fma-def 99.6%

         \[\leadsto \cos th \cdot \frac{\color{blue}{\mathsf{fma}\left(a1, a1, a2 \cdot a2\right)}}{\sqrt{2}} \]

   3. Simplified 99.6%

      \[\leadsto \color{blue}{\cos th \cdot \frac{\mathsf{fma}\left(a1, a1, a2 \cdot a2\right)}{\sqrt{2}}} \]

   4. Taylor expanded in a1 around 0 63.9%

      \[\leadsto \cos th \cdot \color{blue}{\frac{{a2}^{2}}{\sqrt{2}}} \]
   5. Step-by-step derivation

      1. unpow2 63.9%

         \[\leadsto \cos th \cdot \frac{\color{blue}{a2 \cdot a2}}{\sqrt{2}} \]

   6. Simplified 63.9%

      \[\leadsto \cos th \cdot \color{blue}{\frac{a2 \cdot a2}{\sqrt{2}}} \]

   7. Taylor expanded in th around 0 21.1%

      \[\leadsto \color{blue}{1} \cdot \frac{a2 \cdot a2}{\sqrt{2}} \]
   8. Step-by-step derivation

      1. associate-/l* 21.1%

         \[\leadsto 1 \cdot \color{blue}{\frac{a2}{\frac{\sqrt{2}}{a2}}} \]

      2. frac-2neg 21.1%

         \[\leadsto 1 \cdot \color{blue}{\frac{-a2}{-\frac{\sqrt{2}}{a2}}} \]

      3. div-inv 21.1%

         \[\leadsto 1 \cdot \color{blue}{\left(\left(-a2\right) \cdot \frac{1}{-\frac{\sqrt{2}}{a2}}\right)} \]

      4. distribute-neg-frac 21.1%

         \[\leadsto 1 \cdot \left(\left(-a2\right) \cdot \frac{1}{\color{blue}{\frac{-\sqrt{2}}{a2}}}\right) \]

   9. Applied egg-rr 21.1%

      \[\leadsto 1 \cdot \color{blue}{\left(\left(-a2\right) \cdot \frac{1}{\frac{-\sqrt{2}}{a2}}\right)} \]

   11. Applied egg-rr 30.8%

       \[\leadsto 1 \cdot \left(\left(-a2\right) \cdot \frac{1}{\color{blue}{\frac{1}{a2} \cdot \sqrt{2}}}\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 42.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;th \leq 9 \cdot 10^{+61}:\\ \;\;\;\;\frac{a2}{\frac{\sqrt{2}}{a2}}\\ \mathbf{else}:\\ \;\;\;\;a2 \cdot \frac{-1}{\sqrt{2} \cdot \frac{1}{a2}}\\ \end{array} \]

Alternative 7: 39.4% accurate, 3.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;th \leq 9 \cdot 10^{+61}:\\ \;\;\;\;\frac{a2}{\frac{\sqrt{2}}{a2}}\\ \mathbf{else}:\\ \;\;\;\;a2 \cdot \frac{-a2}{\sqrt{2}}\\ \end{array} \end{array} \]

FPCore

(FPCore (a1 a2 th)
 :precision binary64
 (if (<= th 9e+61) (/ a2 (/ (sqrt 2.0) a2)) (* a2 (/ (- a2) (sqrt 2.0)))))

C

double code(double a1, double a2, double th) {
	double tmp;
	if (th <= 9e+61) {
		tmp = a2 / (sqrt(2.0) / a2);
	} else {
		tmp = a2 * (-a2 / sqrt(2.0));
	}
	return tmp;
}

Fortran

real(8) function code(a1, a2, th)
    real(8), intent (in) :: a1
    real(8), intent (in) :: a2
    real(8), intent (in) :: th
    real(8) :: tmp
    if (th <= 9d+61) then
        tmp = a2 / (sqrt(2.0d0) / a2)
    else
        tmp = a2 * (-a2 / sqrt(2.0d0))
    end if
    code = tmp
end function

Java

public static double code(double a1, double a2, double th) {
	double tmp;
	if (th <= 9e+61) {
		tmp = a2 / (Math.sqrt(2.0) / a2);
	} else {
		tmp = a2 * (-a2 / Math.sqrt(2.0));
	}
	return tmp;
}

Python

def code(a1, a2, th):
	tmp = 0
	if th <= 9e+61:
		tmp = a2 / (math.sqrt(2.0) / a2)
	else:
		tmp = a2 * (-a2 / math.sqrt(2.0))
	return tmp

Julia

function code(a1, a2, th)
	tmp = 0.0
	if (th <= 9e+61)
		tmp = Float64(a2 / Float64(sqrt(2.0) / a2));
	else
		tmp = Float64(a2 * Float64(Float64(-a2) / sqrt(2.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a1, a2, th)
	tmp = 0.0;
	if (th <= 9e+61)
		tmp = a2 / (sqrt(2.0) / a2);
	else
		tmp = a2 * (-a2 / sqrt(2.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[a1_, a2_, th_] := If[LessEqual[th, 9e+61], N[(a2 / N[(N[Sqrt[2.0], $MachinePrecision] / a2), $MachinePrecision]), $MachinePrecision], N[(a2 * N[((-a2) / N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if th < 9e61

   1. Initial program 99.6%

      \[\frac{\cos th}{\sqrt{2}} \cdot \left(a1 \cdot a1\right) + \frac{\cos th}{\sqrt{2}} \cdot \left(a2 \cdot a2\right) \]
   2. Step-by-step derivation

      1. distribute-lft-out 99.6%

         \[\leadsto \color{blue}{\frac{\cos th}{\sqrt{2}} \cdot \left(a1 \cdot a1 + a2 \cdot a2\right)} \]

      2. associate-*l/ 99.7%

         \[\leadsto \color{blue}{\frac{\cos th \cdot \left(a1 \cdot a1 + a2 \cdot a2\right)}{\sqrt{2}}} \]

      3. associate-*r/ 99.7%

         \[\leadsto \color{blue}{\cos th \cdot \frac{a1 \cdot a1 + a2 \cdot a2}{\sqrt{2}}} \]

      4. fma-def 99.7%

         \[\leadsto \cos th \cdot \frac{\color{blue}{\mathsf{fma}\left(a1, a1, a2 \cdot a2\right)}}{\sqrt{2}} \]

   3. Simplified 99.7%

      \[\leadsto \color{blue}{\cos th \cdot \frac{\mathsf{fma}\left(a1, a1, a2 \cdot a2\right)}{\sqrt{2}}} \]

   4. Taylor expanded in a1 around 0 59.4%

      \[\leadsto \cos th \cdot \color{blue}{\frac{{a2}^{2}}{\sqrt{2}}} \]
   5. Step-by-step derivation

      1. unpow2 59.4%

         \[\leadsto \cos th \cdot \frac{\color{blue}{a2 \cdot a2}}{\sqrt{2}} \]

   6. Simplified 59.4%

      \[\leadsto \cos th \cdot \color{blue}{\frac{a2 \cdot a2}{\sqrt{2}}} \]
   7. Step-by-step derivation

      1. associate-/l* 59.5%

         \[\leadsto \cos th \cdot \color{blue}{\frac{a2}{\frac{\sqrt{2}}{a2}}} \]

      2. associate-*r/ 59.5%

         \[\leadsto \color{blue}{\frac{\cos th \cdot a2}{\frac{\sqrt{2}}{a2}}} \]

   8. Applied egg-rr 59.5%

      \[\leadsto \color{blue}{\frac{\cos th \cdot a2}{\frac{\sqrt{2}}{a2}}} \]

   9. Taylor expanded in th around 0 45.0%

      \[\leadsto \frac{\color{blue}{a2}}{\frac{\sqrt{2}}{a2}} \]

   if 9e61 < th

   1. Initial program 99.6%

      \[\frac{\cos th}{\sqrt{2}} \cdot \left(a1 \cdot a1\right) + \frac{\cos th}{\sqrt{2}} \cdot \left(a2 \cdot a2\right) \]
   2. Step-by-step derivation

      1. distribute-lft-out 99.6%

         \[\leadsto \color{blue}{\frac{\cos th}{\sqrt{2}} \cdot \left(a1 \cdot a1 + a2 \cdot a2\right)} \]

      2. associate-*l/ 99.6%

         \[\leadsto \color{blue}{\frac{\cos th \cdot \left(a1 \cdot a1 + a2 \cdot a2\right)}{\sqrt{2}}} \]

      3. associate-*r/ 99.6%

         \[\leadsto \color{blue}{\cos th \cdot \frac{a1 \cdot a1 + a2 \cdot a2}{\sqrt{2}}} \]

      4. fma-def 99.6%

         \[\leadsto \cos th \cdot \frac{\color{blue}{\mathsf{fma}\left(a1, a1, a2 \cdot a2\right)}}{\sqrt{2}} \]

   3. Simplified 99.6%

      \[\leadsto \color{blue}{\cos th \cdot \frac{\mathsf{fma}\left(a1, a1, a2 \cdot a2\right)}{\sqrt{2}}} \]

   4. Taylor expanded in a1 around 0 63.9%

      \[\leadsto \cos th \cdot \color{blue}{\frac{{a2}^{2}}{\sqrt{2}}} \]
   5. Step-by-step derivation

      1. unpow2 63.9%

         \[\leadsto \cos th \cdot \frac{\color{blue}{a2 \cdot a2}}{\sqrt{2}} \]

      2. associate-/l* 63.9%

         \[\leadsto \cos th \cdot \color{blue}{\frac{a2}{\frac{\sqrt{2}}{a2}}} \]

   6. Simplified 63.9%

      \[\leadsto \cos th \cdot \color{blue}{\frac{a2}{\frac{\sqrt{2}}{a2}}} \]

   7. Taylor expanded in th around 0 24.8%

      \[\leadsto \color{blue}{\left(1 + -0.5 \cdot {th}^{2}\right)} \cdot \frac{a2}{\frac{\sqrt{2}}{a2}} \]
   8. Step-by-step derivation

      1. unpow2 24.8%

         \[\leadsto \left(1 + -0.5 \cdot \color{blue}{\left(th \cdot th\right)}\right) \cdot \frac{a2}{\frac{\sqrt{2}}{a2}} \]

   9. Simplified 24.8%

      \[\leadsto \color{blue}{\left(1 + -0.5 \cdot \left(th \cdot th\right)\right)} \cdot \frac{a2}{\frac{\sqrt{2}}{a2}} \]

   10. Taylor expanded in th around 0 21.1%

       \[\leadsto \color{blue}{\frac{{a2}^{2}}{\sqrt{2}}} \]
   11. Step-by-step derivation

       1. unpow2 21.1%

          \[\leadsto \frac{\color{blue}{a2 \cdot a2}}{\sqrt{2}} \]

       2. associate-*r/ 21.1%

          \[\leadsto \color{blue}{a2 \cdot \frac{a2}{\sqrt{2}}} \]

   12. Simplified 21.1%

       \[\leadsto \color{blue}{a2 \cdot \frac{a2}{\sqrt{2}}} \]

   14. Applied egg-rr 30.8%

       \[\leadsto \color{blue}{0 - a2 \cdot \frac{a2}{\sqrt{2}}} \]
   15. Step-by-step derivation

       1. neg-sub0 30.8%

          \[\leadsto \color{blue}{-a2 \cdot \frac{a2}{\sqrt{2}}} \]

       2. distribute-lft-neg-in 30.8%

          \[\leadsto \color{blue}{\left(-a2\right) \cdot \frac{a2}{\sqrt{2}}} \]

       3. *-commutative 30.8%

          \[\leadsto \color{blue}{\frac{a2}{\sqrt{2}} \cdot \left(-a2\right)} \]

   16. Simplified 30.8%

       \[\leadsto \color{blue}{\frac{a2}{\sqrt{2}} \cdot \left(-a2\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 42.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;th \leq 9 \cdot 10^{+61}:\\ \;\;\;\;\frac{a2}{\frac{\sqrt{2}}{a2}}\\ \mathbf{else}:\\ \;\;\;\;a2 \cdot \frac{-a2}{\sqrt{2}}\\ \end{array} \]

Alternative 8: 39.3% accurate, 4.0× speedup

Math

\[\begin{array}{l} \\ a2 \cdot \frac{a2}{\sqrt{2}} \end{array} \]

FPCore

(FPCore (a1 a2 th) :precision binary64 (* a2 (/ a2 (sqrt 2.0))))

C

double code(double a1, double a2, double th) {
	return a2 * (a2 / sqrt(2.0));
}

Fortran

real(8) function code(a1, a2, th)
    real(8), intent (in) :: a1
    real(8), intent (in) :: a2
    real(8), intent (in) :: th
    code = a2 * (a2 / sqrt(2.0d0))
end function

Java

public static double code(double a1, double a2, double th) {
	return a2 * (a2 / Math.sqrt(2.0));
}

Python

def code(a1, a2, th):
	return a2 * (a2 / math.sqrt(2.0))

Julia

function code(a1, a2, th)
	return Float64(a2 * Float64(a2 / sqrt(2.0)))
end

MATLAB

function tmp = code(a1, a2, th)
	tmp = a2 * (a2 / sqrt(2.0));
end

Wolfram

code[a1_, a2_, th_] := N[(a2 * N[(a2 / N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.6%

   \[\frac{\cos th}{\sqrt{2}} \cdot \left(a1 \cdot a1\right) + \frac{\cos th}{\sqrt{2}} \cdot \left(a2 \cdot a2\right) \]
2. Step-by-step derivation

   1. distribute-lft-out 99.6%

      \[\leadsto \color{blue}{\frac{\cos th}{\sqrt{2}} \cdot \left(a1 \cdot a1 + a2 \cdot a2\right)} \]

   2. associate-*l/ 99.7%

      \[\leadsto \color{blue}{\frac{\cos th \cdot \left(a1 \cdot a1 + a2 \cdot a2\right)}{\sqrt{2}}} \]

   3. associate-*r/ 99.7%

      \[\leadsto \color{blue}{\cos th \cdot \frac{a1 \cdot a1 + a2 \cdot a2}{\sqrt{2}}} \]

   4. fma-def 99.7%

      \[\leadsto \cos th \cdot \frac{\color{blue}{\mathsf{fma}\left(a1, a1, a2 \cdot a2\right)}}{\sqrt{2}} \]

3. Simplified 99.7%

   \[\leadsto \color{blue}{\cos th \cdot \frac{\mathsf{fma}\left(a1, a1, a2 \cdot a2\right)}{\sqrt{2}}} \]

4. Taylor expanded in a1 around 0 60.3%

   \[\leadsto \cos th \cdot \color{blue}{\frac{{a2}^{2}}{\sqrt{2}}} \]
5. Step-by-step derivation

   1. unpow2 60.3%

      \[\leadsto \cos th \cdot \frac{\color{blue}{a2 \cdot a2}}{\sqrt{2}} \]

   2. associate-/l* 60.4%

      \[\leadsto \cos th \cdot \color{blue}{\frac{a2}{\frac{\sqrt{2}}{a2}}} \]

6. Simplified 60.4%

   \[\leadsto \cos th \cdot \color{blue}{\frac{a2}{\frac{\sqrt{2}}{a2}}} \]

7. Taylor expanded in th around 0 40.0%

   \[\leadsto \color{blue}{\left(1 + -0.5 \cdot {th}^{2}\right)} \cdot \frac{a2}{\frac{\sqrt{2}}{a2}} \]
8. Step-by-step derivation

   1. unpow2 40.0%

      \[\leadsto \left(1 + -0.5 \cdot \color{blue}{\left(th \cdot th\right)}\right) \cdot \frac{a2}{\frac{\sqrt{2}}{a2}} \]

9. Simplified 40.0%

   \[\leadsto \color{blue}{\left(1 + -0.5 \cdot \left(th \cdot th\right)\right)} \cdot \frac{a2}{\frac{\sqrt{2}}{a2}} \]

10. Taylor expanded in th around 0 40.2%

    \[\leadsto \color{blue}{\frac{{a2}^{2}}{\sqrt{2}}} \]
11. Step-by-step derivation

    1. unpow2 40.2%

       \[\leadsto \frac{\color{blue}{a2 \cdot a2}}{\sqrt{2}} \]

    2. associate-*r/ 40.2%

       \[\leadsto \color{blue}{a2 \cdot \frac{a2}{\sqrt{2}}} \]

12. Simplified 40.2%

    \[\leadsto \color{blue}{a2 \cdot \frac{a2}{\sqrt{2}}} \]

13. Final simplification 40.2%

    \[\leadsto a2 \cdot \frac{a2}{\sqrt{2}} \]

Alternative 9: 39.3% accurate, 4.0× speedup

Math

\[\begin{array}{l} \\ \frac{a2}{\frac{\sqrt{2}}{a2}} \end{array} \]

FPCore

(FPCore (a1 a2 th) :precision binary64 (/ a2 (/ (sqrt 2.0) a2)))

C

double code(double a1, double a2, double th) {
	return a2 / (sqrt(2.0) / a2);
}

Fortran

real(8) function code(a1, a2, th)
    real(8), intent (in) :: a1
    real(8), intent (in) :: a2
    real(8), intent (in) :: th
    code = a2 / (sqrt(2.0d0) / a2)
end function

Java

public static double code(double a1, double a2, double th) {
	return a2 / (Math.sqrt(2.0) / a2);
}

Python

def code(a1, a2, th):
	return a2 / (math.sqrt(2.0) / a2)

Julia

function code(a1, a2, th)
	return Float64(a2 / Float64(sqrt(2.0) / a2))
end

MATLAB

function tmp = code(a1, a2, th)
	tmp = a2 / (sqrt(2.0) / a2);
end

Wolfram

code[a1_, a2_, th_] := N[(a2 / N[(N[Sqrt[2.0], $MachinePrecision] / a2), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.6%

   \[\frac{\cos th}{\sqrt{2}} \cdot \left(a1 \cdot a1\right) + \frac{\cos th}{\sqrt{2}} \cdot \left(a2 \cdot a2\right) \]
2. Step-by-step derivation

   1. distribute-lft-out 99.6%

      \[\leadsto \color{blue}{\frac{\cos th}{\sqrt{2}} \cdot \left(a1 \cdot a1 + a2 \cdot a2\right)} \]

   2. associate-*l/ 99.7%

      \[\leadsto \color{blue}{\frac{\cos th \cdot \left(a1 \cdot a1 + a2 \cdot a2\right)}{\sqrt{2}}} \]

   3. associate-*r/ 99.7%

      \[\leadsto \color{blue}{\cos th \cdot \frac{a1 \cdot a1 + a2 \cdot a2}{\sqrt{2}}} \]

   4. fma-def 99.7%

      \[\leadsto \cos th \cdot \frac{\color{blue}{\mathsf{fma}\left(a1, a1, a2 \cdot a2\right)}}{\sqrt{2}} \]

3. Simplified 99.7%

   \[\leadsto \color{blue}{\cos th \cdot \frac{\mathsf{fma}\left(a1, a1, a2 \cdot a2\right)}{\sqrt{2}}} \]

4. Taylor expanded in a1 around 0 60.3%

   \[\leadsto \cos th \cdot \color{blue}{\frac{{a2}^{2}}{\sqrt{2}}} \]
5. Step-by-step derivation

   1. unpow2 60.3%

      \[\leadsto \cos th \cdot \frac{\color{blue}{a2 \cdot a2}}{\sqrt{2}} \]

6. Simplified 60.3%

   \[\leadsto \cos th \cdot \color{blue}{\frac{a2 \cdot a2}{\sqrt{2}}} \]
7. Step-by-step derivation

   1. associate-/l* 60.4%

      \[\leadsto \cos th \cdot \color{blue}{\frac{a2}{\frac{\sqrt{2}}{a2}}} \]

   2. associate-*r/ 60.3%

      \[\leadsto \color{blue}{\frac{\cos th \cdot a2}{\frac{\sqrt{2}}{a2}}} \]

8. Applied egg-rr 60.3%

   \[\leadsto \color{blue}{\frac{\cos th \cdot a2}{\frac{\sqrt{2}}{a2}}} \]

9. Taylor expanded in th around 0 40.2%

   \[\leadsto \frac{\color{blue}{a2}}{\frac{\sqrt{2}}{a2}} \]

10. Final simplification 40.2%

    \[\leadsto \frac{a2}{\frac{\sqrt{2}}{a2}} \]

Reproduce

herbie shell --seed 2023250 
(FPCore (a1 a2 th)
  :name "Migdal et al, Equation (64)"
  :precision binary64
  (+ (* (/ (cos th) (sqrt 2.0)) (* a1 a1)) (* (/ (cos th) (sqrt 2.0)) (* a2 a2))))